Application of neural networks to turbulence control for drag reduction

Transcription

1 Application of neural networks to turbulence control for drag reduction Changhoon Lee and John Kim a) Department of Mechanical and Aerospace Engineering, University of California at Los Angeles, Los Angeles, California David Babcock and Rodney Goodman Department of Electrical Engineering, California Institute of Technology, Pasadena, California Received 16 September 1996; accepted 24 February 1997 A new adaptive controller based on a neural network was constructed and applied to turbulent channel flow for drag reduction. A simple control network, which employs blowing and suction at the wall based only on the wall-shear stresses in the spanwise direction, was shown to reduce the skin friction by as much as 20% in direct numerical simulations of a low-reynolds number turbulent channel flow. Also, a stable pattern was observed in the distribution of weights associated with the neural network. This allowed us to derive a simple control scheme that produced the same amount of drag reduction. This simple control scheme generates optimum wall blowing and suction proportional to a local sum of the wall-shear stress in the spanwise direction. The distribution of corresponding weights is simple and localized, thus making real implementation relatively easy. Turbulence characteristics and relevant practical issues are also discussed American Institute of Physics. S I. INTRODUCTION The ability to control turbulent flows is of significant economic interest. Successful control of turbulent boundary layers by reducing drag, for example, can result in a substantial reduction in operational cost for commercial aircraft and marine vehicles. Recent studies 1,2 show that near-wall streamwise vortices are responsible for high skin-friction drag in turbulent boundary layers. Some attempts have been made to reduce the skin-friction drag by controlling the interactions between these vortices and the wall. Choi et al., 2 for example, used blowing and suction at the wall equal and opposite to the wall-normal component of velocity at y 10. They showed that this control effectively mitigated the streamwise vortices, giving approximately 25% drag reduction in a turbulent channel flow. Although the method employed in their work is impractical, since the information at y 10 is usually not available, it demonstrates a control scheme which reduces skin-friction drag by manipulation of the near-wall streamwise vortices. Another way to control the streamwise vortices is to impose spanwise oscillation by a moving wall or externally imposed body force. 3,4 These methods, however, require a large amount of energy input. A systematic approach using suboptimal control theory has also been tried in the past. This approach, which attempts to minimize a cost function, was successfully applied to the stochastic Burgers equation. 5 Moin and Bewley 6 applied a similar approach to a turbulent channel flow to achieve up to 50% drag reduction. The control, however, requires information from the entire velocity field inside the flow domain and excessive computational time, making it impractical to implement in real situations. For practical implementation, a control scheme should utilize only quantities that are easily a Corresponding author: Telephone: ; Fax: ; Electronic mail: jkim@turb.seas.ucla.edu measurable at the wall, and should be fast enough to be applied in real time. The objective of the present work is to seek wall actuations, in the form of blowing and suction at the wall, dependent on the wall-shear stress to achieve a substantial skinfriction reduction. This requires knowledge of how the wallshear stresses respond to wall actuations, i.e., the correlation between the wall-shear stresses and the wall actuations. Because of the complexity of the solutions to the Navier Stokes equations, however, it is not possible to find such a correlation in closed form or to approximate it in simple form. Instead, we use a neural network to approximate the correlation which then predicts the optimal wall actuations to achieve the minimum value of the skin-friction drag. Neural networks have been used to obtain complicated, nonlinear correlations without a priori knowledge of the system that is to be controlled. Jacobson and Reynolds, 7 for instance, used a neural network to obtain about 7% drag reduction in their simulation of an artificial flow. In this paper, we describe how we constructed and trained a neural network off-line, and then implemented an on-line control scheme blowing and suction at the wall for drag reduction based on that neural network. Applying this control scheme to direct numerical simulations of turbulent channel flow at low Reynolds number, we observed about 20% drag reduction. We then describe how examination of the weight distribution from the on-line neural network led to a very simple control scheme that worked equally well while being computationally more efficient. In Sec. II, a brief description of the architecture of the neural network used in the present work is given. In Sec. III, results obtained from control using an off-line trained network are presented, while results obtained from control using a network with continuous on-line training are given in Sec. IV. In Sec. V, a simple control law based on the weight distribution from the successful neural network control is presented. A few turbulence statistics are given in Sec. VI, 1740 Phys. Fluids 9 (6), June /97/9(6)/1740/8/$ American Institute of Physics

2 domain grid points in each direction. The summation is done over the spanwise direction. Seven neighboring points (N 7), including the point of interest, in the spanwise direction corresponding to approximately 90 wall units with our numerical resolution were found to provide enough information to adequately train and control the near-wall structures responsible for the high-skin friction. Note that the blowing and suction are applied at each grid location according to 1 as a numerical approximation of distributed blowing and suction on the surface. A scaled conjugate gradient learning algorithm 8 was used to produce rapid training. For given pairs of (v des jk, w/ y jk ), network was trained to minimize the sum of a weighted-squared error given by FIG. 1. Neural network architecture. error 1 2 e v des jk v des jk v net jk 2, j k 2 followed in Sec. VII by a discussion of practical implementation and the conclusions. In this paper we use (x,y,z) for the streamwise, wallnormal, and spanwise coordinates, respectively, and (u,v,w) for the corresponding velocity components. II. NEURAL NETWORK In this section we describe the construction of a neural network to learn the correlation between wall-shear stresses and wall actuations from a given data set. Although a neural network generally requires no prior knowledge of the system or plant, knowledge about the near-wall turbulence structures provides a guideline for the design of the network architecture. Initially u/ y and w/ y at the wall at several instances of time were used as input data fields and the actuation at the wall was used for the output data of the network. Experimentally we found that only w/ y at the wall from the current instance of time was necessary for sufficient network performance. Because we wanted the output to be based only on a local input area, we designed our network using shared weights. The network had a single set of weights a template that is convolved over the entire input space to generate output values; that is, we used the same set of weights for each data point and the training involves iterating over all data points. The template extracts spatially invariant correlations between input and output data. The size of the template was initially chosen to include information about a single streak and streamwise vortex, and then was varied to find an optimal size. We used a standard two-layer feedforward network with hyperbolic tangent hidden units and a linear output unit see Fig. 1. The functional form of our final neural network is N 1 /2 w v jk W a tanh b W i W W c, i N 1 /2 y j,k i 1 j N x, 1 k N z, 1 where the W s denote weights, N is the total number of input weights, and the subscripts j and k denote the numerical grid point at the wall in the streamwise and spanwise directions respectively. N x and N z are the number of computational where v des is the desired output value and v net is the network output value given by Eq. 1. The weights were initialized with a set of random numbers. Note that the error defined in 2 exponentially emphasizes proportional to ) large actuations. This error scaling was chosen based on Choi et al. s 2 observation that large actuations are more important for drag reduction. Usually within 100 training epochs, the error reached its asymptotic limit. III. OFF-LINE TRAINING AND CONTROL As an initial experiment we investigated whether we could train a neural network to predict the velocity at y 10 from only the wall-shear stresses. The rationale behind this experiment was to duplicate an existing control law using only measurements available at the wall such that the output from the network could be used as input to the actuator. The network should yield a similar amount of drag reduction to that obtained by Choi et al. 2 through prediction of the velocity at y 10. The training data consisted of 100 time steps of output obtained from a numerical simulation of channel flow under the control employed by Choi et al., 2 i.e., using the wall-normal velocity at y 10. The flow regime is turbulent channel flow with Re 100, where Re is the Reynolds number based on the wall-shear velocity, u, and the channel half-width,. All numerical simulations presented in this paper were obtained using a modified version of Kim et al. s 9 spectral code with the computational domain (4,2,4 /3), and a grid resolution of 32, 65, 32 in the (x, y,z) directions, respectively. Each time step contained a array of input values ( w/ y w ) and corresponding actuations, v at y 10. We trained several networks with one hidden unit and different sized input templates: 7 1, 7 3, 7 5, 9 1, 9 3, 11 1, and 11 3 the number of input points in the spanwise direction by the number of input points in the streamwise direction. After training was completed, the distribution of input weights was examined to see whether there was a discernible pattern in the input template. The weight distributions in the spanwise direction at the same streamwise location for different input templates are shown in Fig. 2. The same pattern for all 7 input template sizes was observed, and is similar to a finite difference approximation of Phys. Fluids, Vol. 9, No. 6, June 1997 Lee et al. 1741

3 FIG. 2. Weight distribution from off-line trainings for various sizes of the input template: 7 1, 7 3, 7 5, 9 1, 9 3, 11 1, Weights are normalized by W 1. FIG. 3. Mean wall-shear stress histories for various control laws compared to the no-control case:, no control;, control with 7 fixed weights obtained from off-line control;, control with 10 fixed weights from the same off-line control. All runs with control shown in this paper were integrated over at least 20 time units that is much longer than the typical turnover time of the streamwise vortices, 1 time unit. FIG. 4. Schematic representation of adaptive inverse model control. spanwise differentiation. Increasing the number of hidden units only marginally improved performance, whereas increasing the template size significantly reduced the final training error. We then applied a control scheme to a regular channel flow by fixing the final input weights obtained from the offline training. This was perhaps a somewhat naive approach since the weights were obtained from fully controlled flow data and the shear stresses used for the training were already altered by the actuations. Nevertheless, two cases were tested: a control scheme based on 7 weights in the spanwise direction, and another based on the same 7 weights plus 3 more in the immediate downstream location. These 10 weights were chosen because among all weights obtained from the off-line training they had non-negligible values. In Fig. 3, the mean shear stress variations at the wall i.e., drag obtained with these two controls are plotted along with the no-control case. Nearly 18% drag reduction was achieved, with slightly better performance from the 10-weight network. These results demonstrate that a correlation exists between the shear stresses at the wall and the desired actuations, and that control based on this correlation produces a significant amount of drag reduction. This fixed-weight control scheme, however, was deduced from fully controlled flow data; thus it does not guarantee the same performance for other flow situations. IV. ON-LINE CONTROL In the previous section we showed how successful control based on off-line training can be obtained. However, since the system we are trying to control is time-varying and nonlinear, this approach is not likely to generalize well. Continuous on-line training allows a controller to adapt to the evolution of the system. In this section, we describe an adaptive controller for on-line training and control. There are various schemes for on-line neural network control. The most direct scheme is adaptive-inverse model control. 10 A schematic representation of this approach is shown in Fig. 4. Here the plant denotes the numerical solver of the Navier Stokes equations. This configuration employs a neural network to model the possibly time-varying inverse plant mapping from wall-shear stress w/ y w to the wall-normal actuations, and then uses a copy of the model as the controller with the desired shear stresses as input. One restriction of this technique is that it usually requires an initial model training phase using random plant inputs and corresponding plant outputs. This, however, caused no serious problems since usually one time step was enough for model training because of the shared-weight architecture in our application each time is 1024 data points and our networks only have a few weights. Once the model represents a reasonably close approximation to the actual plant inverse, a copy is then implemented as a feedforward controller. The desired inputs to the controller are a fractional reduction in the shear stress from the previous step, i.e., w des y w y, 3 t t t where 0 1. Based on the networks and techniques we investigated, this indirect suppression of w/ y w, instead of u/ y w, turns out to be more efficient in achieving drag reduction. The output of the controller, which is the input to the plant, is the predicted actuation necessary to produce this shear-stress reduction. The quantity should be chosen such 1742 Phys. Fluids, Vol. 9, No. 6, June 1997 Lee et al.

4 FIG. 5. Distribution of the weights:, from on-line training;, A(1 cos( j))/j. z 13. that sets of the desired large amplitude outputs are among the sets that are well represented by the training sets. Good performance was achieved for the range of A turbulent channel flow at Reynolds number Re 100 was used to test the neural network. We allowed all the weights in the network to adapt and examined the input template pattern after each time step. As the control began, the weight distribution immediately assumed a fixed pattern similar to the off-line pattern. There was no appreciable change in the relative magnitudes of the template weights over time, indicating that the pattern is preserved. The absolute magnitudes, however, did vary indicating the need for gain and bias adaptation for each layer. The number of hidden layers, the number of the hidden units, the size of the input template, and the error scale of the training error of Eq. 2 were 1, 1, 7 1, and 5, respectively. We varied the values of the error scale and found 5 to be optimum, with larger values causing an instability in training. We also varied the number of the hidden units, but the network with a single hidden unit produced the best result. The converged weight distributions for 20 consecutive time steps after an asymptotic state was reached are shown in Fig. 5. The pattern is only slightly different from the one obtained from the off-line training see Fig. 2. It should be noted that this pattern emerged immediately after the on-line control began. Time histories of the wall-shear stress for 3 different input template sizes are shown in Fig. 6. All other network parameters are kept the same. The abscissa represents nondimensional time normalized by u and. One time unit roughly corresponds to the time for a fluid particle traveling with the centerline velocity to move about 20 channel half widths. The computations were carried out long enough to ensure that a statistically steady state was reached. As the control began, the drag quickly drops to about 80% of that observed without control, for the two cases with template sizes of 7 1 and 9 1. The template size of 5 1, however, did not produce as much reduction, indicating that at least 7 spanwise points which extended about 90 wall units, with our grid resolution should be used for good performance. At the initial stage of our study, we tested a control using both w/ y w and u/ y w as input, with a 7 5 input template. It produced about the same amount of reduction, but with excessive training time due to a significantly larger number FIG. 6. Mean wall-shear stress histories for on-line control with different input template sizes:, no control;, control with 5 1;, control with 7 1;, control with 9 1. of weights. Furthermore, it did not produce coherent patterns in the weight distributions. Since the 7 weights showed a fixed pattern, we fixed the input template weights to this pattern and used a single hidden unit network, giving only 4 adaptable parameters a bias and gain for each layer. This simplified network had the following functional form: v jk W a tanh W b g W c W d 4 with 3 w g i 3 W i, 5 y j,k i where W i s are the fixed-weight pattern obtained from the previous on-line control. On-line control using this network produced a similar amount of drag reduction. The weight variations with time were also monitored. The bias weights (W c and W d ) were negligibly small. Those controlling the gain (W a and W b ), which had finite values, changed in time significantly, although the product of the two gain-weights remained almost constant. This suggests that effective control can be achieved by simply using g with an adjustable amplitude. This will be discussed in the following section. V. A SIMPLE CONTROL SCHEME Since the control based on the network given by Eq. 4 produced a substantial reduction in drag, this section develops a control scheme based only on the weighted sum of the wall-shear stress. The distribution of the weights can be approximated by see Fig. 5 : 1 cos j W j A, 6 j where j 0 corresponds to the point where the control is applied. Since only the relative values are important, the constant A has no special meaning. A computation with a twice higher resolution was run to confirm that Eq. 6 gives the proper form of the weight distribution see Fig. 7. It turns out that Eq. 6 is the inverse Fourier transform of ik z / k z, where k z is the wave number in the spanwise direction, for the finite maximum wave number, i.e., Phys. Fluids, Vol. 9, No. 6, June 1997 Lee et al. 1743

5 FIG. 7. Distribution of the weights:, from on-line training;, A(1 cos( j))/j. z 6. FIG. 9. Mean wall-shear stress histories for various control schemes compared to the no-control case:, no control;, on-line control with neural network with 7 1 template; control with 7 fixed weights. k m ik z km k z exp ik zz dk z 2 1 cos k mz, 7 z where k m / z is the maximum wave number and z is the numerical grid spacing. Replacing z with j z in the right-hand-side of Eq. 7 leads to Eq. 6. From this result and the convolution theorem, one can suggest the following simple control law for wall transpiration: vˆ ik z w ˆ w C k z y w C 1 k z w ˆ z y, where the hat denotes a Fourier transformed quantity and C is a positive scale factor determining the amplitude of the actuation. Equation 8, which should produce similar drag reduction as the result with 7 fixed weights Eq. 4, implies that the optimum blowing and suction at the wall is proportional to ( w/ y)/ z with the high wave number component suppressed by 1/ k z. Note that the wall-normal actuation proportional to ( w/ y)/ z counteracts the up-anddown motion induced by a streamwise vortex see Fig. 8, consistent with that used by Choi et al. 2 Smith et al. 11 discussed the stability of the near-wall turbulence structure to a local adverse pressure gradient imposed near the surface. FIG. 8. A schematic of a flow field induced by a streamwise vortex and corresponding ( w/ y w )/ z. 8 The suction at the left side of the vortex in Fig. 8 can relieve such adverse pressure gradient, thus preventing the surfacelayer eruption. Choi et al. 2 reported that actuation based only on the local value of ( w/ y)/ z, which is also a dominant term of the Taylor expansion of v at some distance away from the wall, did not produce such a substantial drag reduction. This indicates that suppression of high wave number components by 1/ k z, or equivalently using locally weighted value of w/ y, plays a key role in reducing drag. Note that the weights at even numbered grid points away from the center point vanish. This is beneficial for physical implementation, since sensors and actuators cannot be placed at the same location. 12 Because the Fourier integral is computed for a finite value of k m, the values of the weight at noninteger j in Eq. 6 have no meaning. Equation 8 is equivalent to N 1 /2 w v jk C W i, 9 i N 1 /2 y j,k i where W i is given by Eq. 6. The magnitude of the weights decays with increasing distance from the center, which allows for good approximation using only a small number of weights, i.e., successful control requires only local values of the shear stress. A natural concern is how different grid spacings affect the control. Since the above control law Eq. 9 is simply another expression of Eq. 8, which is a good approximation as long as k m is large enough, the control should be relatively independent of resolution. This was confirmed by a computation with a higher resolution. Control based on Eq. 9 with 7 points produced the same amount of drag reduction 20% as the neural network control see Fig. 9. The constant C is chosen so that the root-mean-squared rms value of the actuation is kept at 0.15u. Blowing and suction of this magnitude at the wall suppress the near-wall streamwise vortices by counteracting the up-and-down motions associated with these vortices see Fig. 8. The fact that a control scheme that is essentially linear Eq. 9 produced the same amount of drag reduction as on-line control with a nonlinear network led to the following question: Can the same weight pattern be captured when on-line control with a linear network is used? To answer 1744 Phys. Fluids, Vol. 9, No. 6, June 1997 Lee et al.

6 FIG. 11. Probability-density function of the wall-shear stress:, no control;, control with 7 fixed weights. The area under each curve is normalized to one. FIG. 10. Contours of streamwise vorticity in a cross-flow plane: a no control; b control using 7 fixed weights. The contour level increment is the same for both figures. Negative contours are chain-dotted. this question, we tested an on-line controller utilizing a linear network. The hyperbolic tangent function in Eq. 1 was replaced with a linear function and the hidden layer was removed. All other parameters were kept the same as the nonlinear network case. The linear network seemed to produce almost the same pattern in weights as the one produced by our nonlinear network. Also, drag was reduced by the same amount. The pattern that was obtained by averaging over time, however, was not well preserved in time. The weight W 3 of Eq. 1, for example, frequently showed order of magnitude excursions resulting in a large standard deviation, 1.5, compared to the value of the weight, 0.3 weights were normalized by the maximum amplitude W 1 ). The nonlinear network, however, produced more constant weights; the standard deviation of W 3 was only This result indicates that the nonlinearity should add robustness by bounding the weights. This can simplify hardware implementation by limiting the signals and subsequently the necessary dynamic range of the processing circuitry. The computed flow fields for a no-control case and a successful control case, based on the 7-point weighted sum of w/ y w Eq. 9, were examined to investigate the mechanism by which the drag reduction is achieved. The most salient feature of the controlled case was that the strength of the near-wall streamwise vortices was drastically reduced. In Fig. 10, contours of streamwise vorticity in a cross plane are shown. This result further substantiates the notion that a successful suppression of the near-wall streamwise vortices leads to a significant reduction in drag. Note that for the controlled case the wall actuations were applied at both walls. The probability-density function of the wall-shear stress in the streamwise direction is shown in Fig. 11. It is evident that the control is very effective in suppressing large fluctuations, thus reducing the mean-skin friction. Furthermore, the root-mean-square rms values of turbulent fluctuations in the wall region are also reduced as shown in Fig. 12 a. The same trend was observed by Choi et al. 2 The rms vorticity fluctuations are also significantly reduced, except for x very VI. TURBULENCE STATISTICS FIG. 12. Root-mean-square fluctuations normalized by the wall variables:, no control;, control with 7 fixed weights. a Velocity fluctuations; b vorticity fluctuations. Phys. Fluids, Vol. 9, No. 6, June 1997 Lee et al. 1745

7 FIG. 13. Contours of the wall actuation: a control using w/ y w with 7 fixed weights; b control using information at y 10. Negative contours are chain-dotted. close to the wall. The increase for the latter is caused by additional v/ z due to the wall actuation. The rms fluctuations of z at the wall, which is mainly due to u/ y at the wall, is also decreased. This indicates that the control scheme led to a reduction in the mean shear and its variance at the wall by suppressing large fluctuations as shown in Fig. 11. The reduction in the rms fluctuations in x and y indicates that the control scheme indeed reduced the strength of the near-wall streamwise vortices and the wall-layer streaks. Figure 13 compares the distribution of wall actuation used in our control with that from Choi et al. s 2 v-control using the information at y 10 for the same wall-shear stress distribution. The corresponding wall-shear stress distribution is also shown in Fig. 13. The wall actuations indicate a strikingly similar distribution to each other, even though the wall actuation of our control is based only on the wall-shear stress w/ y w. Basically our control scheme detects edges of locally high-shear stress regions by measuring the spanwise variation of w/ y, and applies appropriate wall actuation as shown in Figs. 13 a and 13 b. Since highshear stress regions are usually elongated in the streamwise direction, only spanwise variation is necessary for detecting the edges. Since w/ y is a direct measurement of streamwise vortices see Fig. 8, it provides more appropriate information than u/ y. This is consistent with our finding that w/ y at several points in spanwise direction are enough for good performance of control. VII. DISCUSSION AND CONCLUSION We have presented a successful application of a neural network to turbulence control for drag reduction. First we were able to construct and train a neural network off-line to find a correlation between the wall-shear stress and the desired wall actuations coming from the information at y 10. Based on the optimal network structure from the off-line training, we successfully implemented an on-line inverse model controller in numerical experiments of a turbulent channel flow, resulting in about 20% drag reduction. Finally, we were able to deduce a simple control scheme Eq. 9 for drag reduction based on observation of the weight distribution from a successful control case. This control scheme uses a minimal amount of wall-shear stress information and requires only simple operations, thus rendering a scheme whose actual hardware implementation would be relatively easy. Beginning with a nonlinear network, we found a simple linear control scheme from the pattern of the weights, suggesting the use of a linear network. Although a linear network produced a similar pattern, the pattern was not well preserved in time. The weights can vary significantly increasing the difficulty in circuit implementation due to the larger required dynamic range. The nonlinear network, however, produced bounded weights simplifying hardware implementation. This suggests that a certain amount of nonlinearity is beneficial to capture a stable coherent pattern for our system. Although the present control scheme is a significant improvement over earlier approaches that require velocity information within the flow, there is a technical issue before the present control scheme can be implemented in real practice. Precise control of blowing and suction distributed over a surface is difficult to implement. Other approaches, such as surface movements by deformable surface, may prove to be more practical. There are also several fundamental issues that must be addressed. First, our numerical experiments were performed in a very low Reynolds number flow. It remains to be seen whether the same control scheme extends to higher Reynolds numbers. If the main cause of high-skin friction in turbulent boundary layers at higher Reynolds numbers is also due to the near-wall streamwise vortices, the same scheme should work equally well. Detection of these vortices through the wall-shear stress would become more difficult, however, because the scales associated with these vortices decrease as the Reynolds number increases. Another important issue is the influence of the spatial resolution of sensors and actuators on performance. We showed that a simple control scheme with 4 neighboring points in the spanwise direction 3 among 7 weights are zeros, which span 90 wall units, performed very well. This suggests that the distance between sensors should be about 20 wall units. As the Reynolds number increases, the physical separation between sensors must decrease. We note in passing that recent advances in micromachined sensors and actuators make these scales feasible, 13 and the present work is part of a joint research project aimed at integrating micronsized sensors and actuators with analog VLSI control circuitry. A third issue is determining an appropriate scale factor C in Eq. 9. We tested a range of C and empirically found 1746 Phys. Fluids, Vol. 9, No. 6, June 1997 Lee et al.

8 that values yielding an actuation rms value between 0.1u and 0.15u gave the best performance. Smaller values resulted in less reduction, while larger ones caused rapid fluctuations of the wall-shear stress over time. The experiments were only performed for Re 100 and 180, from which the optimum value was deduced, but we expect that the same amplitude level should produce similar reductions for higher Reynolds number flows. We further note that for this amplitude the required power input per unit area to produce the actuation, p w v w 0.5 v w 3 ( 0.1 u 3 ) was significantly less compared to the power saved due to the reduced drag, C f /C f w U c ( 3.5 u 3 ) for 20% reduction in C f. Here p w,, C f, w and U c are the wall pressure, density, friction coefficient, averaged wall-shear stress in the streamwise direction, and centerline velocity, respectively. Note that we did not account for device loss to deliver blowing and suction nor the power consumed by sensors in this estimate. One final practical issue worth mentioning is the time delay between sensing and actuation. None was included in any of our numerical experiments. In a real situation, however, there will be a finite time delay between sensing and actuation. The process time for the simple control scheme should be negligible since a weighted summation is an involved process, thus not imposing any restriction on the time delay. Ideally, sensors should register the response of the near-wall structures to be controlled due to wall actuations, which will take a certain amount of time. Blowing and suction at the wall, however, can produce an immediate spurious response at nearby sensors due to local conservation. One possible remedy to this problem is an under-relaxation of the actuation signal with past signals, accounting for the response of a longer time scale. For example, an underrelaxation scheme using the following formula v t t jk C i W i w t y j,k i t 1 v jk 10 could take into account all past signals with a single parameter (0 1). We used this approach successfully in our numerical experiment for Re 180 and found that there is an optimum depending on temporal resolution. The most significant finding of the present study is that a single spanwise strip of w/ y w is enough to achieve significant drag reduction. Additional information about w/ y w in the streamwise direction or u/ y w in either direction reduced the efficiency of our neural-network based control. We also tested different-sized input templates and found that as the template size increased in the streamwise direction, the training time became excessive and the fixed pattern in the weight disappeared, indicating that too many unnecessary weights caused training problems. ACKNOWLEDGMENTS We are grateful to Dr. Gary Coleman for comments on a draft of this paper and Tim Berger for preparation of Figs. 10 and 13. We also are grateful to a referee for suggestion about the linear network. This work is supported by AFOSR Grants Nos. F and F Dr. James M. McMichael, Program Manager. It is also supported in part by the Center for Neuromorphic Systems Engineering as a part of the National Science Foundation Engineering Research Center Program under Grant No. EEC Computer time has been supplied by the San Diego Supercomputer Center, NASA Ames Research Center and by the NAS program at NASA Ames Research Center. 1 J. Kim, Study of turbulence structure through numerical simulations: The perspective of drag reduction, AGARD Report No. R-786, AGARD FDP/VKI Special Course on Skin Friction Drag Reduction, 2 6 March 1992, VKI, Brussels H. Choi, P. Moin, and J. Kim, Active turbulence control for drag reduction in wall-bounded flows, J. Fluid Mech. 262, R. Akhavan, W. J. Jung, and N. Mangiavacchi, Turbulence control in wall-bounded flows by spanwise oscillations, Appl. Sci. Res. 51, J. Kim, C. Lee, T. Berger, J. Lim, and H. Choi, Effects of electromagnetic force on near-wall turbulence, Bull. Am. Phys. Soc. 40, H. Choi, R. Temam, P. Moin, and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech. 253, P. Moin and T. Bewley, Application of control theory to turbulence, in the 12th Australasian Fluid Mechanics Conference, Sydney December 1995 unpublished, p S. L. Jacobson and W. C. Reynolds, Active control of boundary layer wall shear stress using self-learning neural networks, AIAA Shear Flow Conference, Orlando, 1993 unpublished, p.1. 8 M. Moller, Efficient training of feed-forward neural networks, Ph.D. thesis, Aarhus University, Denmark, J. Kim, P. Moin, and R. Moser, Turbulence statistics in fully-developed channel flow at low Reynolds number, J. Fluid Mech. 177, B. Widrow, Adaptive inverse control, In the Second IFAC Workshop on Adaptive Systems in Control and Signal Processing, Lund, Sweden, 1987 unpublished, pp C. R. Smith, J. D. A. Walker, A. H. Haidari, and U. Sobrun, On the dynamics of near-wall turbulence, Philos. Trans. R. Soc. London Ser. A 336, This staggered distribution of sensors and actuators would result in actuators at only half the grid points and this might reduce the effectiveness of the control. This problem can be solved simply by increasing resolution twice so that actuators locate at every grid point of the old grid system these correspond to half the grid points in the new grid system still with sensors located in staggered positions. This is possible since the weight distribution does not depend on the grid distance, but depends on the grid index as shown in Figs. 5 and C. Liu, Y. C. Tai, J. B. Huang, and C. M. 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